Technically they could, but it would mean creating and managing a new license fee system.
Right now there's no enforcement of the license fee, just the knowledge that, to watch live broadcasts (either on TV or online) you must have a license, and if a TV license inspector catches you, you'll be in trouble.
With the iPlayer service, if you watch any of the live-streaming channels and you don't have a TV License you will eventually get a letter informing you that you need to pay the license fee. Fail to obtain the TV License after that point and you are subject to the normal fines.
I think the BBC must have some form of tie-in with the ISPs as I know a number of people that have had this happen to them.
I'm regularly tunnelling all my traffic through a UK VPS (not for geolocation regions, I'm in the UK myself anyway), and never had any issues watching iPlayer.
And if I watched to avoid it, I could always use something like www.tvcatchup.com to watch the live content, and just go onto the iPlayer for VOD content (for which you don't need a license).
I wonder - traditionally a TV license is per household, not per person, meaning that if I want to watch on a TV in a friend's house, they need to have a license, not me. With internet access, how does that work. For example, say they were able to prove I had been watching iPlayer live streams in my house, what if it was a friend watching on their laptop, and they own a license? What if you're watching from a public WiFi or a 3G connection on a train? The "license for the house" doesn't quite translate.
Right now there's no enforcement of the license fee, just the knowledge that, to watch live broadcasts (either on TV or online) you must have a license, and if a TV license inspector catches you, you'll be in trouble.